nLab horizontal categorification

Idea

Horizontal categorification or oidification (to be contrasted with the more famous vertical categorification which is usually just called categorification) describes the process by which

1. a concept is realized to be equivalent to a certain type of category or magmoid with a single object;

2. and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.

Examples

Example

(groups to groupoids)
The horizontal categorification of groups are groupoids: categories in which every morphism is invertible.

Similar comments as the following Remark apply generally to oidification:

Remark

In Ex. there is an “inclusion” 2-functor of (the 1-category Grp) of groups into (the (2,1)-category Grpd) of groupoids, by passage to delooping groupoids $\mathbf{B}(-)$,

$$\begin{array}{ccc} Grp &\longrightarrow& Grpd \\ G &\mapsto& \mathbf{B}G \mathrlap{\,.} \end{array}$$

Beware, though, that this 2-functor is not fully faithful: For $G, H \in Grpd$, the hom-groupoid between their delooping groupoids is not quite the set of group homomorphisms $G \to H$, but is the action groupoid of that by the pointwise adjoint action of $H$:

$$Grpd\big( \mathbf{B}G ,\, \mathbf{B}H \big) \;\simeq\; Grp\big( G ,\, H \big) \sslash_{ad} H \mathrlap{\,.}$$

For instance, if $H = A$ is abelian then

$$Grpd\big( \mathbf{B}G ,\, \mathbf{B}A \big) \;\simeq\; Grp\big( G ,\, A \big) \times \mathbf{B}A \mathrlap{\,.}$$

This is one formal way of seeing that it is not quite right to say that “groupoids are generalized groups”, not without further qualification:

Namely, since $\mathbf{B}G$ is canonically a pointed object in Grpd (uniquely pointed by its single object), one may instead consider the delooping-construction to land in the (2,1)-category $Grpd^\ast$ of pointed groupoids

$$\begin{array}{ccc} Grp &\longrightarrow& Grpd^\ast \\ G &\mapsto& \mathbf{B}G \end{array}$$

and as such this is fully faithful, equivalently identifying groups with pointed connected groupoids:

$$Grpd^\ast\big( \mathbf{B}G ,\, \mathbf{B}H \big) \;\simeq\; Hom\big( G ,\, H \big) \mathrlap{\,.}$$

This is part of a general phenomenon discussed further at looping and delooping (cf. also at May recognition theorem).

In practice, for instance Picard groupoids appear as generalized abelian groups: Indeed, Picard groupoids are abelian 2-groups and as such are in particular pointed (by their neutral element, the tensor unit-object).

Example

(algebras to algebroids)
A horizontal categorification of algebras are algebroids: categories enriched in the category of vector spaces, regarded as a symmetric monoidal category via the tensor product of vector spaces.

Example

(rings to ringoids)
A horizontal categorification of rings are ringoids: categories enriched over the category of abelian groups, via the tensor product of abelian groups (cf. blog).

Example

($C^\ast$-algebras to $C^\ast$-categories)
A horizontal categorification of $C^\*$-algebras hence ought to be known as $C^\ast$–algebroids but is usually known as $C^\ast$-categories.

Example

(groups to groupoids)
Since, by the Gelfand-Naimark theorem, $C^\ast$-algebras are dual to topological spaces, Paolo Bertozzini et. al proposed to define spaceoids to be the entities formally dual to $C^*$-categories (cf. blog).

Example

(monoids to categories)
And an exception to the rule: The many-object verion of monoids are not called a monoidoids – but are called… categories!

On the other hand, in the case of enrichment over R Mod (for $R$ a commutative ring), monoids are known as associative algebras over $R$ and the oidificies $R Mod$-categories may reasonably be and sometimes are called algebroid over $R$, as in Ex.

Example

(monads to bicategory-enriched categories)
A horizontal categorification of the notion of monads is that of categories enriched in a bicategory.

Example

(Lawvere theories and operads)
A horizontal categorification of a single-sorted Lawvere theory are multisorted Lawvere theory. Analogously operads categorify to a many-colored operad.

Example

In type theory/programming language theory, horizontal categorification is analogous to introducing a type distinction: an untyped language is a special case of a typed language where there is exactly one type. This fits with the categorical semantics: untyped lambda calculus has models in Lawvere theories, whereas (simply) typed lambda calculus has models in Cartesian multicategories.

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

General definition

Tom Leinster’s book “Higher Operads, Higher Categories” contains a general theory of horizontal categorification of a notion of algebraic structure, as long as the algebraic structure can be defined as the category of algebras for an operad. Leinster proves that if $P$ is a (non-symmetric) operad in the category of sets, then $P$ can be extended to an “fc-multicategory” $\Sigma P$, the suspension of $P$, which is a kind of generalized operad which acts on directed graphs. So, for example, the horizontal categorification of the algebraic theory of monoids, as an operad, is the algebraic theory of categories, as a generalized operad. Eugenia Cheng has extended this definition of suspension so that it works for non-symmetric operads in a monoidal category $\mathcal{V}$ satisfying certain co/completeness assumptions. For example, one can take $\mathcal{V}= \mathbf{Cat}$. In this case, the operad $P$ whose algebras are monoidal categories has for its suspension (i.e., its horizontal categorification) the generalized operad whose algebras are precisely bicategories.

Further discussion

• John Baez, What is categorification?

• John Baez, Ringoids

• John Baez, Higher Operads, Higher Categories

• Eugenia Cheng, Comparing operadic theories of $n$-category